Coexistence of triple, quadruple attractors and Wada basin boundaries in a predator-prey model with additional food for predators. (English) Zbl 1512.92067
Summary: In biological control programs, the provision of additional food is regarded as one of the most useful tools for species conservation and pest management. In a system, providing additional food for predators can distract the predators from the overconsumption of prey (short term) or enhance the predation rate (long term). In the present article, we consider a predator-prey model with Holling type-II functional response by incorporating additional food for the predator species in a discrete-time setup. Here, we intend to explore the impact of additional food on the growth of prey as well as the overall dynamics of the underlying system. We analyze the system dynamics by varying two control parameters, viz, the biomass of additional food and the intrinsic growth rate of prey species simultaneously with the help of Lyapunov exponent and isoperiodic diagrams. Through the simultaneous variation of two parameters, very rich and complex dynamical behaviors of a simple deterministic system can come into view with the presence of structurally stable periodic patterns in the transitional and chaotic zones, which is not looked on only by the variability of one parameter. In this study, we notice the presence of infinite families of different organized periodic structures, like Arnold tongues, shrimp-shaped domains, a new kind of ‘double fishhook’ structures, etc. in the quasiperiodic and chaotic zones of the parameter planes with various kinds of period-adding phenomena. The study also discloses the transition to chaos via shrimp-induced period-bubbling process. We observe the coexistence of double and triple attractors in several different sets of attractors. The most fascinating finding in this study is the coexistence of quadruple attractors, one of the rarest phenomena in ecological systems. The basin boundaries of all these coexisting double, triple, and quadruple attractors are of a very complex nature, they are either fractal basin boundaries or Wada basin boundaries.
Keywords:
additional food; Arnold tongues; shrimp-shaped structures; triple attractor; quadruple attractor; Wada basinReferences:
| [1] | Roberts, John A. G.; Quispel, Gilles R. W., Chaos and time-reversal symmetry. Order and chaos in reversible dynamical systems, Phys Rep, 216, 2-3, 63-177 (1992) |
| [2] | Gallas, Jason A. C., Structure of the parameter space of the Hénon map, Phys Rev Lett, 70, 18, 2714 (1993) |
| [3] | Façanha, Wilson; Oldeman, Bart; Glass, Leon, Bifurcation structures in two-dimensional maps: The endoskeletons of shrimps, Phys Lett A, 377, 18, 1264-1268 (2013) · Zbl 1290.37026 |
| [4] | Mugnaine, Michele; Sales, Matheus R.; Szezech Jr., José D.; Viana, Ricardo L., Dynamics, multistability, and crisis analysis of a sine-circle nontwist map, Phys Rev E, 106, 3, Article 034203 pp. (2022) |
| [5] | Hossain, Mainul; Kumbhakar, Ruma; Pal, Nikhil, Dynamics in the biparametric spaces of a three-species food chain model with vigilance, Chaos Solitons Fractals, 162, Article 112438 pp. (2022) · Zbl 1506.92073 |
| [6] | de Souza, Silvio L. T.; Lima, Angélica A.; Caldas, Iberê L.; Medrano-T, Rene O.; Guimarães-Filho, Zwinglio de O., Self-similarities of periodic structures for a discrete model of a two-gene system, Phys Lett A, 376, 15, 1290-1294 (2012) · Zbl 1260.37060 |
| [7] | Rössler, Otto E.; Letellier, Christophe, Chaos: The world of nonperiodic oscillations (2020), Springer Nature, Switzerland · Zbl 1439.34003 |
| [8] | De Oliveira, Juliano A.; Montero, Leonardo T.; Da Costa, Diogo R.; Méndez-Bermúdez, J. A.; Medrano-T, Rene O.; Leonel, Edson D., An investigation of the parameter space for a family of dissipative mappings, Chaos, 29, 5, Article 053114 pp. (2019) |
| [9] | Garai, Shilpa; Pati, N. C.; Pal, Nikhil; Layek, G. C., Organized periodic structures and coexistence of triple attractors in a predator-prey model with fear and refuge, Chaos Solitons Fractals, 165, Article 112833 pp. (2022) · Zbl 1508.92208 |
| [10] | Bonatto, Cristian; Gallas, Jason A. C., Accumulation boundaries: codimension-two accumulation of accumulations in phase diagrams of semiconductor lasers, electric circuits, atmospheric and chemical oscillators, Phil Trans R Soc A, 366, 1865, 505-517 (2008) · Zbl 1153.78324 |
| [11] | Hossain, Mainul; Garai, Shilpa; Jafari, Sajad; Pal, Nikhil, Bifurcation, chaos, multistability, and organized structures in a predator-prey model with vigilance, Chaos, 32, 6, Article 063139 pp. (2022) · Zbl 07874290 |
| [12] | Pati, N. C.; Garai, Shilpa; Hossain, Mainul; Layek, G. C.; Pal, Nikhil, Fear induced multistability in a predator-prey model, Int J Bifurcation Chaos, 31, 10, Article 2150150 pp. (2021) · Zbl 1475.37103 |
| [13] | Medeiros, Everton S.; De Souza, Silvio L. T.; Medrano-T, Rene O.; Caldas, Iberêl L., Replicate periodic windows in the parameter space of driven oscillators, Chaos Solitons Fractals, 44, 11, 982-989 (2011) |
| [14] | Perez, Rafael; Glass, Leon, Bistability, period doubling bifurcations and chaos in a periodically forced oscillator, Phys Lett A, 90, 9, 441-443 (1982) |
| [15] | Glass, Leon; Guevara, Michael R.; Belair, Jacques; Shrier, Alvin, Global bifurcations of a periodically forced biological oscillator, Phys Rev A, 29, 3, 1348 (1984) |
| [16] | Hilborn, Robert C., Chaos and nonlinear dynamics: An introduction for scientists and engineers (2000), Oxford University Press: Oxford University Press New York · Zbl 1015.37001 |
| [17] | Glass, Leon; Perez, Rafael, Fine structure of phase locking, Phys Rev Lett, 48, 26, 1772 (1982) |
| [18] | Arnold, Vladimir I., Small denominators, I: mappings of the circumference into itself, Am Math Soc Transl: Ser 2, 46, 213 (1965) · Zbl 0152.41905 |
| [19] | Fraser, Simon; Kapral, Raymond, Analysis of flow hysteresis by a one-dimensional map, Phys Rev A, 25, 6, 3223 (1982) |
| [20] | Gaspard, Pierre; Kapral, Raymond; Nicolis, Grégoire, Bifurcation phenomena near homoclinic systems: A two-parameter analysis, J Stat Phys, 35, 5, 697-727 (1984) · Zbl 0588.58055 |
| [21] | Bélair, Jacques; Glass, Leon, Universality and self-similarity in the bifurcations of circle maps, Physica D, 16, 2, 143-154 (1985) · Zbl 0593.58030 |
| [22] | Gallas, Jason A. C., Dissecting shrimps: results for some one-dimensional physical models, Phys A, 202, 1, 196-223 (1994) |
| [23] | Oliveira, Diego F. M.; Robnik, Marko; Leonel, Edson D., Shrimp-shape domains in a dissipative kicked rotator, Chaos, 21, 4, Article 043122 pp. (2011) |
| [24] | Milnor, John, Remarks on iterated cubic maps, Experiment Math, 1, 1, 5-24 (1992) · Zbl 0762.58018 |
| [25] | Carcasses, Jean P.; Mira, Christian; Bosch, M.; Simó, C.; Tatjer, Joan C., “Crossroad area-spring area” transition (i) parameter plane representation, Int J Bifurcation Chaos, 1, 1, 183-196 (1991) · Zbl 0758.58025 |
| [26] | Lorenz, Edward N., Compound windows of the Hénon-map, Physica D, 237, 13, 1689-1704 (2008) · Zbl 1155.37030 |
| [27] | Bonatto, Cristian; Garreau, Jean C.; Gallas, Jason A. C., Self-similarities in the frequency-amplitude space of a loss-modulated CO \({}_2\) laser, Phys Rev Lett, 95, 14, Article 143905 pp. (2005) |
| [28] | Stoop, Ruedi; Benner, Philipp; Uwate, Yoko, Real-world existence and origins of the spiral organization of shrimp-shaped domains, Phys Rev Lett, 105, 7, Article 074102 pp. (2010) |
| [29] | Seydel, Rüdiger, Practical bifurcation and stability analysis, vol. 5 (2009), Springer Science & Business Media, New York · Zbl 1195.34004 |
| [30] | Rajagopal, Karthikeyan; Jafari, Sajad; Pham, Viet-Thanh; Wei, Zhouchao; Premraj, Durairaj; Thamilmaran, Kathamuthu; Karthikeyan, Anitha, Antimonotonicity, bifurcation and multistability in the Vallis model for El Niño, Int J Bifurcation Chaos, 29, 3, Article 1950032 pp. (2019) · Zbl 1414.34037 |
| [31] | Brzeski, Piotr; Lazarek, Mateusz; Kapitaniak, Tomasz; Kurths, Jürgen; Perlikowski, Przemysław, Basin stability approach for quantifying responses of multistable systems with parameters mismatch, Meccanica, 51, 11, 2713-2726 (2016) |
| [32] | Schooler, Shon S.; Salau, Buck; Julien, Mic H.; Ives, Anthony R., Alternative stable states explain unpredictable biological control of Salvinia molesta in Kakadu, Nature, 470, 7332, 86-89 (2011) |
| [33] | Attneave, Fred, Multistability in perception, Sci Am, 225, 6, 62-71 (1971) |
| [34] | Arecchi, Fortunato T.; Meucci, Riccardo; Puccioni, G.; Tredicce, J., Experimental evidence of subharmonic bifurcations, multistability, and turbulence in a Q-switched gas laser, Phys Rev Lett, 49, 17, 1217 (1982) |
| [35] | Lenderink, Geert; Haarsma, Reindert J., Variability and multiple equilibria of the thermohaline circulation associated with deep-water formation, J Phys Oceanogr, 24, 7, 1480-1493 (1994) |
| [36] | Shiau, Yuo-Hsien; Peng, Yih-Ferng; Hwang, Robert R.; Hu, Chin-Kun, Multistability and symmetry breaking in the two-dimensional flow around a square cylinder, Phys Rev E, 60, 5, 6188 (1999) |
| [37] | Xie, Jierui; Sreenivasan, Sameet; Korniss, Gyorgy; Zhang, Weituo; Lim, Chjan; Szymanski, Boleslaw K., Social consensus through the influence of committed minorities, Phys Rev E, 84, 1, Article 011130 pp. (2011) |
| [38] | Ullner, Ekkehard; Zaikin, Alexei; Volkov, Evgenii I.; García-Ojalvo, Jordi, Multistability and clustering in a population of synthetic genetic oscillators via phase-repulsive cell-to-cell communication, Phys Rev Lett, 99, 14, Article 148103 pp. (2007) |
| [39] | Ullner, Ekkehard; Koseska, Aneta; Kurths, Jürgen; Volkov, Evgenii; Kantz, Holger; García-Ojalvo, Jordi, Multistability of synthetic genetic networks with repressive cell-to-cell communication, Phys Rev E, 78, 3, Article 031904 pp. (2008) |
| [40] | Pisarchik, Alexander N.; Feudel, Ulrike, Control of multistability, Phys Rep, 540, 4, 167-218 (2014) · Zbl 1357.34105 |
| [41] | Laurent, Michel; Kellershohn, Nicolas, Multistability: a major means of differentiation and evolution in biological systems, Trends Biochem Sci, 24, 11, 418-422 (1999) |
| [42] | Seekell, David A.; Cline, Timothy J.; Carpenter, Stephen R.; Pace, Michael L., Evidence of alternate attractors from a whole-ecosystem regime shift experiment, Theor Ecol, 6, 3, 385-394 (2013) |
| [43] | Dercole, Fabio; Ferrière, Régis; Rinaldi, Sergio, Ecological bistability and evolutionary reversals under asymmetrical competition, Evolution, 56, 6, 1081-1090 (2002) |
| [44] | Schröder, Arne; Persson, Lennart; De Roos, André M., Direct experimental evidence for alternative stable states: a review, Oikos, 110, 1, 3-19 (2005) |
| [45] | Srinivasu, P. D.N.; Prasad, B. S.R. V.; Venkatesulu, M., Biological control through provision of additional food to predators: a theoretical study, Theor Popul Biol, 72, 1, 111-120 (2007) · Zbl 1123.92039 |
| [46] | Elkinton, Joseph S.; Liebhold, Andrew M.; Muzika, Rose-Marie, Effects of alternative prey on predation by small mammals on gypsy moth pupae, Popul Ecol, 46, 2, 171-178 (2004) |
| [47] | Zhu, Yan-lan; Zhou, Wei; Chu, Tong; Elsadany, A. A., Complex dynamical behavior and numerical simulation of a Cournot-Bertrand duopoly game with heterogeneous players, Commun Nonlinear Sci Numer Simul, 101, Article 105898 pp. (2021) · Zbl 1468.91081 |
| [48] | Srinivasu, P. D.N.; Prasad, B. S.R. V., Time optimal control of an additional food provided predator– prey system with applications to pest management and biological conservation, J Math Biol, 60, 4, 591-613 (2010) · Zbl 1311.92171 |
| [49] | Layek, G. C., An introduction to dynamical systems and chaos (2015), Springer, New Delhi · Zbl 1354.34001 |
| [50] | Strogatz, Steven H., Nonlinear dynamics and chaos: With applications to physics, biology, chemistry, and engineering (2018), CRC Press: CRC Press New York · Zbl 1343.37001 |
| [51] | Benettin, Giancarlo; Galgani, Luigi; Giorgilli, Antonio; Strelcyn, Jean-Marie, Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Part 1: Theory, Meccanica, 15, 1, 9-20 (1980) · Zbl 0488.70015 |
| [52] | Schuster, Heinz G.; Just, Wolfram, Deterministic chaos: An introduction (2006), John Wiley & Sons,Weinheim · Zbl 1094.37001 |
| [53] | Feigenbaum, Mitchell J., Quantitative universality for a class of nonlinear transformations, J Stat Phys, 19, 1, 25-52 (1978) · Zbl 0509.58037 |
| [54] | Bier, Martin; Bountis, Tassos C., Remerging Feigenbaum trees in dynamical systems, Phys Lett A, 104, 5, 239-244 (1984) |
| [55] | Dawson, Silvina P.; Grebogi, Celso; Yorke, James A.; Kan, Ittai; Koçak, Hüseyin, Antimonotonicity: inevitable reversals of period-doubling cascades, Phys Lett A, 162, 3, 249-254 (1992) |
| [56] | Parlitz, Ulrich; Lauterborn, Werner, Superstructure in the bifurcation set of the Duffing equation \(\ddot{x} + d \dot{x} + x + x^3 = f c o s ( \omega t )\), Phys Lett A, 107, 8, 351-355 (1985) |
| [57] | Parlitz, Ulrich; Lauterborn, Werner, Period-doubling cascades and devil’s staircases of the driven Van der Pol oscillator, Phys Rev A, 36, 3, 1428 (1987) |
| [58] | Kocarev, Ljupco; Halle, Sean; Eckert, Kevin; Chua, Leon O., Experimental observation of antimonotonicity in Chua’s circuit, Int J Bifurcation Chaos, 3, 4, 1051-1055 (1993) · Zbl 0894.58061 |
| [59] | Vandermeer, John, Period ‘bubbling’ in simple ecological models: pattern and chaos formation in a quartic model, Ecol Model, 95, 2-3, 311-317 (1997) |
| [60] | McDonald, Steven W.; Grebogi, Celso; Ott, Edward; Yorke, James A., Fractal basin boundaries, Physica D, 17, 2, 125-153 (1985) · Zbl 0588.58033 |
| [61] | Lai, Ying-Cheng; Tél, Tamás, Transient chaos: Complex dynamics on finite time scales (2011), Springer Science & Business Media: Springer Science & Business Media New York · Zbl 1218.37005 |
| [62] | Mandelbrot, Benoit, Fractals (1977), Freeman: Freeman San Francisco · Zbl 0376.28020 |
| [63] | Barnsley, Michael F., Fractals everywhere (2014), Academic Press: Academic Press New York · Zbl 0691.58001 |
| [64] | Losa, Gabrielle A.; Baumann, Gerd; Nonnenmacher, Theo F., Fractal dimension of pericellular membranes in human lymphocytes and lymphoblastic leukemia cells, Pathol-Res Pract, 188, 4-5, 680-686 (1992) |
| [65] | Brú, Antonio; Albertos, Sonia; Subiza, José L.; García-Asenjo, José L.; Brú, Isabel, The universal dynamics of tumor growth, Biophys J, 85, 5, 2948-2961 (2003) |
| [66] | Katz, A. J.; Thompson, A. H., Fractal sandstone pores: implications for conductivity and pore formation, Phys Rev Lett, 54, 12, 1325 (1985) |
| [67] | Burrough, Peter A., Fractal dimensions of landscapes and other environmental data, Nature, 294, 5838, 240-242 (1981) |
| [68] | De Moura, Alessandro P. S.; Grebogi, Celso, Output functions and fractal dimensions in dynamical systems, Phys Rev Lett, 86, 13, 2778 (2001) |
| [69] | Aguirre, Jacobo; Sanjuán, Miguel A. F., Unpredictable behavior in the Duffing oscillator: Wada basins, Physica D, 171, 1-2, 41-51 (2002) · Zbl 1008.37011 |
| [70] | Vandermeer, John; Stone, Lewi; Blasius, Bernd, Categories of chaos and fractal basin boundaries in forced predator-prey models, Chaos Solitons Fractals, 12, 2, 265-276 (2001) · Zbl 0976.92033 |
| [71] | Gong, Pu-Lin; Xu, Jian-Xue, Global dynamics and stochastic resonance of the forced FitzHugh-Nagumo neuron model, Phys Rev E, 63, 3, Article 031906 pp. (2001) |
| [72] | Daza, Alvar; Wagemakers, Alexandre; Sanjuán, Miguel A. F., Ascertaining when a basin is Wada: the merging method, Sci Rep, 8, 1, 1-8 (2018) |
| [73] | Yoneyama, Kunizô, Theory of continuous set of points (not finished), Tohoku Math J, First Ser, 12, 43-158 (1917) · JFM 46.0303.01 |
| [74] | Kennedy, Judy; Yorke, James A., Basins of wada, Physica D, 51, 1-3, 213-225 (1991) · Zbl 0746.58054 |
| [75] | Aguirre, Jacobo; Vallejo, Juan C.; Sanjuán, Miguel A. F., Wada basins and chaotic invariant sets in the Hénon-Heiles system, Phys Rev E, 64, 6, Article 066208 pp. (2001) |
| [76] | Nusse, Helena E.; Yorke, James A., Wada basin boundaries and basin cells, Physica D, 90, 3, 242-261 (1996) · Zbl 0886.58072 |
| [77] | Aguirre, Jacobo; Vallejo, Juan C.; Sanjuan, Miguel A. F., Wada basins and unpredictability in Hamiltonian and dissipative systems, Internat J Modern Phys B, 17, 22-24, 4171-4175 (2003) · Zbl 1073.37024 |
| [78] | Daza, Alvar; Wagemakers, Alexandre; Georgeot, Bertrand; Guéry-Odelin, David; Sanjuán, Miguel A. F., Basin entropy: a new tool to analyze uncertainty in dynamical systems, Sci Rep, 6, 1, 1-10 (2016) |
| [79] | Poon, Leon; Campos, José; Ott, Edward; Grebogi, Celso, Wada basin boundaries in chaotic scattering, Int J Bifurcation Chaos, 6, 2, 251-265 (1996) · Zbl 0870.58069 |
| [80] | Zhang, Yongxiang; Luo, Guanwei; Cao, Qingjie; Lin, Mei, Wada basin dynamics of a shallow arch oscillator with more than 20 coexisting low-period periodic attractors, Int J Non-Linear Mech, 58, 151-161 (2014) |
| [81] | Zhang, Yongxiang; Zhang, Huaguang; Gao, Wenzhong, Multiple Wada basins with common boundaries in nonlinear driven oscillators, Nonlinear Dynam, 79, 4, 2667-2674 (2015) |
| [82] | Daza, Alvar; Wagemakers, Alexandre; Sanjuán, Miguel A. F., Wada property in systems with delay, Commun Nonlinear Sci Numer Simul, 43, 220-226 (2017) · Zbl 1469.34085 |
| [83] | Daza, Alvar; Shipley, Jake O.; Dolan, Sam R.; Sanjuán, Miguel A. F., Wada structures in a binary black hole system, Phys Rev D, 98, 8, Article 084050 pp. (2018) |
| [84] | Vandermeer, John, Wada basins and qualitative unpredictability in ecological models: a graphical interpretation, Ecol Model, 176, 1-2, 65-74 (2004) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
